Question

PN is any chord of the parabola y2 =4ax; the point M divides PN in the ratio m:n.find the locus of M.

Answer 1

Answer on Question #47278 – Math – Analytic Geometry

PN is any chord of the parabola $y^{2} = 4ax$; the point M divides PN in the ratio m:n. find the locus of M.

Solution:

We consider point P on the parabola and N on the x-axis. Let $(x_{1},y_{1})$ be the point on the parabola, and M will be (x, y). Then we can write the following:

$$y _ {1} ^ {2} = 4 a x _ {1}$$

Then we can note the following equity accordingly to the condition of the task:

$$x = x _ {1}$$ $$y = \frac {y _ {1} n}{(m + n)}$$

Based on the first formula we can find the square of the y.

$$y ^ {2} = \frac {y _ {1} ^ {2} n ^ {2}}{(m + n) ^ {2}}$$

Now we can eliminate the $y_1^2$ from the first and last equations. We obtained the following result.

$$y ^ {2} = \frac {4 a x _ {1} n ^ {2}}{(m + n) ^ {2}}$$

We apply the second equation to eliminate the $x_{1}$. We obtained the resulting equation.

$$y ^ {2} = \frac {4 a x n ^ {2}}{(m + n) ^ {2}}$$

We can rewrite the noted above equation.

$$y ^ {2} = 4 \left[ \frac {a n ^ {2}}{(m + n) ^ {2}} \right] x$$ $$4 a x n ^ {2} = y ^ {2} (m + n) ^ {2}$$

The locus is the parabola $y^{2} = 4bx$ where:

$$b = \frac {y ^ {2}}{4 x} = \frac {4 a x n ^ {2}}{(m + n) ^ {2}} \div \frac {4 x}{1} = \frac {4 a x n ^ {2}}{(m + n) ^ {2}} \cdot \frac {1}{4 x}$$

Simplify the equation. Finally we obtained the find value.

$$b = \frac {a n ^ {2}}{(m + n) ^ {2}}$$

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